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The Radius of Convergence of a Power Series Examples 1

Recall from The Radius of Convergence of a Power Series page that we can calculate the radius of convergence of a power series using the ratio test, that is if $\lim_{n \to \infty} \biggr \rvert \frac{a_{n+1}}{a_n} \biggr \rvert = L$, then the radius of convergence is $R = \frac{1}{L}$. If $L = 0$ then the radius of convergence $R = \infty$ and if $L = \infty$ then the radius of convergence $R = 0$.

We will now look at some more examples of determining the radius of convergence of a given power series.

Example 1

Determine the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$.

We first notice that $a_n = \frac{1}{n!}$ in our power series, and applying the rule above we have that